User:Tohline/AxisymmetricConfigurations/SolvingPE

Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions

You have arrived at this page from our Tiled Menu by clicking on the chapter title that is also referenced in the panel of the following table that is colored light blue. You may proceed directly to that chapter by clicking (again) on the same chapter title, as it appears in the table. However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected. The common thread is the "Key Equation" presented in the top panel of the table.

Synopses

The gravitational potential (both inside and outside) of any axisymmetric mass distribution may be determined from the integral expression, <math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math> <math>~=</math> <math>~ - \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(\varpi~ \varpi^')^{1 / 2}} \biggr] K(\mu) \rho(\varpi^', z^') 2\pi \varpi^'~ d\varpi^' dz^' </math> <math>\mathrm{where:}~~~\mu \equiv \{4\varpi \varpi^' /[ (\varpi+\varpi^')^2 + (z-z^')^2]\}^{1 / 2}</math> where, <math>~K(\mu)</math> is the complete elliptic integral of the first kind. This "Key Equation" may be straightforwardly obtained, for example, by combining Eqs. (31), (32b), and (24) from Cohl & Tohline (1999) and recognizing that the relevant differential area, <math>~d\sigma^' = \varpi^' d\varpi^' dz^' \int_0^{2\pi} d\varphi = 2\pi\varpi^'~d\varpi^' dz^'</math>; see also, Bannikova et al. (2011), Trova, Huré & Hersant (2012), and Fukushima (2016).

Dyson-Wong Tori (Thin Ring Approximation) In §102 of a book titled, The Theory of the Potential, W. D. MacMillan (1958; originally, 1930) derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>. Throughout our related discussions, we generally will refer to this "Key Equation" from MacMillan as providing an expression for the, Gravitational Potential in the Thin Ring (TR) Approximation <math>~\Phi_\mathrm{TR}(\varpi,z)</math> <math>~=</math> <math>~-\biggl[ \frac{2GM}{\pi } \biggr]\frac{K(k)}{\sqrt{(\varpi+a)^2 + z^2}}</math> <math>\mathrm{where:}~~~k \equiv \{4\varpi a/[ (\varpi+a)^2 + z^2]\}^{1 / 2}</math> See also, O. D. Kellogg (1929), §III.4, Exercise (4). As is reviewed in the chapter of our H_Book titled, Dyson-Wong Tori, a number of research groups over the years have re-derived this "thin ring" approximation in the context of their search for effective and insightful ways to determine the gravitational potential of axisymmetric systems.

Solving the Poisson Equation Solving the Poisson Equation You have arrived at this page from our Tiled Menu by clicking on the chapter title that is also referenced in the panel of the following table that is colored light blue. You may proceed directly to that chapter by clicking (again) on the same chapter title, as it appears in the table. However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected. The common thread is the "Key Equation" presented in the top panel of the table.

Using Toroidal Coordinates to Determine the Gravitational Potential Using Toroidal Coordinates to Determine the Gravitational Potential

Wong's (1973) Analytic Potential Wong's (1973) Analytic Potential

Trova, Huré & Hersant (2012)

Related Discussions

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